Ah HA! Some internet searching turned up several references to Rev. Rul. 62-77.

Revenue Ruling 63-77: The IRS specifies that reimbursements made to individuals by a prospective employer for expenses incurred in connection with travel for interviews are not considered wages for federal tax purposes.

It's been my experience that when I'm tempted to say, "A-ha!" I'm about to get into trouble. Alas, this affliction seems to affect you too. First, read the entire ruling, not just a summary to which you attach the implications you desire:

"Wages" is a term of art, and it has nothing to do with what the prospective employer did here. Wages are paid by employers to employees, and what that ruling says that under specific circumstances monies paid on behalf of prospective employees are not wages paid to an employee for employment tax purposes. Nothing the 1099 issuer has done is contrary to this ruling.

I seriously doubt that your husband had exactly $1,000 in expenses which he meticulously documented to the payor. Rather, it sounds like they figured his cost to be roughly $1,000 including a little something for his time and trouble and cut him a check in that amount. Such an arrangement with a nonemployee is correctly reported as nonemployee compensation on Form 1099-MISC. Your husband reports it on Schedule C along with the expenses he can document. The net is subject to Self-Employment Tax.